The stress-energy tensor of an Unruh-DeWitt detector

Daniel A. T. Vanzella; J. P. M. Pitelli; T. Rick Perche

REVIEW 4 major objections 6 minor 1 cited by

The stress-energy tensor of an Unruh-DeWitt detector

T0 review · 4 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A finite-size Unruh-DeWitt detector can carry a covariantly conserved stress-energy tensor that satisfies all four standard energy conditions when its localization is modeled dynamically.

desk verdict Useful construction of a covariant detector stress-energy tensor, but the 'very general conditions' claim overreaches beyond the one engineered example. read the letter →

arxiv 2411.09732 v2 pith:K5M4GY3K submitted 2024-11-14 quant-ph gr-qchep-th

T. Rick Perche , J. P. M. Pitelli , Daniel A. T. Vanzella This is my paper · ORCID

classification quant-phgr-qchep-th

keywords Unruh-DeWittdetectorstress-energytensorenergyconditionsperfectfluidlocalizedquantumfieldcovariantLagrangianbackreactionparticle

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proposes a model of a finite-size particle detector whose stress-energy tensor can actually be computed and trusted in a relativistic setting. The authors argue that a detector's localization must come from dynamical fields rather than a prescribed external potential, because a prescribed potential breaks general covariance and leaves the energy tensor unconserved. Their Lagrangian couples the detector field $\phi_d$ to a complex scalar $\psi_c$ and a perfect fluid, and they show that the resulting total stress-energy tensor is covariantly conserved when the fluid satisfies a specific differential condition. In a concrete spherically symmetric example they verify that the renormalized detector tensor obeys the null, weak, strong, and dominant energy conditions. The payoff is a particle-detector model that can serve as a physically reasonable source for gravitational backreaction.

What carries the argument

The load-bearing object is the full Lagrangian $L = -\frac{1}{2}\partial_\mu\phi_d\partial^\mu\phi_d - \frac{m_d^2}{2}\phi_d^2 - \frac{\alpha}{2}|\psi_c|^2\phi_d^2 - \partial_\mu\psi_c^*\partial^\mu\psi_c - m_c^2|\psi_c|^2 - V_c(|\psi_c|^2) + (1-\mu|\psi_c|^2)L_{\text{fluid}}$ of Eq. (20). The complex scalar $\psi_c$ supplies the confining potential for the detector field, and the non-minimal coupling to the perfect fluid lets the fluid's on-shell Lagrangian participate in the field equations and stabilize the profile of $\psi_c$. The argument turns on the divergence identity in Eqs. (26)-(27): the total tensor is conserved exactly when $(1-\mu|\psi_c|^2)\partial_\mu T^{\text{fluid}\,\mu\nu} - \mu T^{\text{fluid}}_{\mu\nu}\partial^\mu|\psi_c|^2 + \mu L_{\text{fluid}}\partial^\nu|\psi_c|^2 = 0$, which reduces to an ordinary differential equation for the pressure in the static, spherically symmetric example. The fluid's two degrees of freedom are what let the configuration absorb changes in the detector state while staying covariant.

What would settle it

Take a detector state beyond the ground and first excited states, for instance a spatial profile $g(x)$ from a different bound-mode shape or a superposition, solve the fluid pressure equation (34), and check whether a solution with $\rho>0$, $\rho+P>0$, $\rho+3P>0$, and $\rho-|P|>0$ exists for some $\mu \in (0,\ell^2)$; if no such solution exists, the claim that the fluid can absorb arbitrary backreaction while preserving the energy conditions is false.

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Extended reading notes

Core claim

The paper's central claim is that a UDW detector can be described covariantly by the Lagrangian of Eq. (20), with the detector field $\phi_d$ localized by a potential generated by the complex field $\psi_c$, whose profile is stabilized by a perfect fluid. For this Lagrangian the total stress-energy tensor (25) is conserved on shell provided the fluid satisfies Eq. (27), a differential condition on its velocity, energy density, and pressure. In the explicit example with $\psi_c(x) = \ell^{-1}e^{-i\omega_c t}\operatorname{sech}(r/\ell)$, $\alpha = -6$, and $\mu$ in the range $0 < \mu < \ell^2/(1+(1-3\eta)g_0/2)$ (for $\eta = 0$, $\mu \lesssim 0.565\,\ell^2$), the renormalized detector stress-energy tensor takes the diagonal form (62) with distinct radial and angular pressures, and the authors show it satisfies all four standard energy conditions. The result is a concrete demonstration that the energy tensor of a detector plus its localization machinery can be a physically reasonable, covariantly conserved object.

Load-bearing premise

The construction assumes that a perfect fluid can always be adjusted to absorb the detector's backreaction while still behaving as non-exotic matter, but the paper only demonstrates this adjustment for two states of a single spherically symmetric example.

Editorial extensions

If this is right

A particle detector described this way has a covariantly conserved stress-energy tensor, so it can appear as a source in semiclassical gravity without an external-potential inconsistency.
The detector's renormalized energy tensor in the explicit model satisfies the null, weak, strong, and dominant energy conditions, meaning the detector plus its localization system is made of non-exotic matter.
When the detector is excited, the fluid's pressure and density adjust to absorb the backreaction, and the total tensor remains in the same diagonal form, so the construction can follow the detector through a measurement process.
In the pointlike limit the model reproduces the excitation probability of a standard Unruh-DeWitt detector, so it reduces to the familiar detector phenomenology while adding a consistent energy tensor.
The stress-energy tensor of the detector's final state after a measurement is a mixture of the ground- and excited-state tensors weighted by the excitation probability, giving a concrete operational prediction for the energy cost of a measurement.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: the same fluid-absorption mechanism should generalize to other localized probes, such as graviton detectors or multiple entangled detectors, whenever their localization can be written as a potential generated by a classical field.
Editorial inference: extending the model to curved spacetimes would let one compute the gravitational backreaction of a UDW detector order by order, but the paper does not carry out that extension.
Editorial inference: a numerical scan over detector states beyond the two shown, or over non-spherically-symmetric profiles, would test the 'very general conditions' claim; the paper leaves that as an open question.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Read the Perche–Pitelli–Vanzella paper on the detector stress-energy tensor. What's genuinely new: instead of putting the confining potential for the detector field in by hand, they source it from a complex scalar field whose profile is pinned by a perfect fluid. That gives a Lagrangian with a covariantly conserved stress-energy tensor. The derivation in Eqs. (25)-(27) is standard and the conservation check is clean. The explicit example in Sec. V is worked out in real detail: sech profile, closed-form P(0), plots of the energy conditions, pressure deviator, and a pointlike limit check on the excitation probability. This is a useful proof of principle.

Where I'd push back: the abstract says the tensor satisfies the energy conditions "under very general conditions," but the demonstration is one spherically symmetric example with a specific sech profile, a specific self-interaction, alpha = -6, and a fluid Lagrangian tuned so that Eq. (34) can be solved. The fluid absorbs the state dependence of g(x) by construction; Eq. (70) makes the on-shell fluid Lagrangian explicitly state-dependent. No general existence argument is given for arbitrary backreaction, so the "very general" phrasing is an extrapolation. Also, the excited-state tensor components rho1, R1, P1 are only plotted, not written out; the remark that the expressions are cumbersome doesn't help a referee verify the energy-condition claim independently. That is a minor but real reproducibility gap.

The physical status of the fluid is acceptable as a designed model, but the state-dependent on-shell Lagrangian should be flagged more prominently as a feature rather than passed over.

Bottom line: this deserves a serious referee. The construction is explicit and the main derivation is sound. I would ask the authors to weaken the "very general" claim, provide the excited-state expressions (or a supplementary notebook), and prove the smoothness and positivity of P(r) over the full claimed range of mu. After those revisions, it is a solid contribution to relativistic quantum information.

Referee Report

4 major / 6 minor

Summary. The paper constructs a semiclassical model of a finite-size Unruh-DeWitt detector: a real scalar field ϕd is localized by coupling to a complex scalar ψc, which is in turn localized by a perfect fluid with a non-minimal coupling. The authors derive the total stress-energy tensor from the covariant Lagrangian in Eq. (20), show that on-shell conservation is equivalent to the fluid equation (27), and then construct an explicit spherically symmetric example with a sech profile, α=-6, and a specific fluid Lagrangian. For this example they obtain an exactly solvable bound mode for ϕd, compute the renormalized detector stress-energy tensor in the ground and first excited states, and claim that the null, weak, strong, and dominant energy conditions are satisfied. The paper is best read as a proof-of-principle construction rather than a general theorem: the 'very general conditions' of the abstract are supported only by one family of examples, with several analytic gaps and plot-based checks.

Significance. If fully established, the model would be a valuable step toward covariantly consistent descriptions of localized particle detectors and their gravitational backreaction. The paper's strengths include a clean derivation of the stress-energy tensor and conservation condition from a fixed Lagrangian, an exact localized mode for the detector field, a comparison with the standard UDW excitation probability, and the observation that the resulting matter is not a perfect fluid via the pressure deviator. However, the advertised generality of the energy-condition result currently exceeds the analytic content: the general existence of physical fluid configurations for arbitrary detector states is not proved, and the energy-condition verification relies on integrals, numerical plots, and unstated monotonicity/inequality facts. The central construction is sound as a designed example, but the manuscript needs either substantially stronger proofs or a more carefully calibrated statement of what is demonstrated.

major comments (4)

[Abstract; Sec. IV, Eqs. (34)-(35)] The abstract claims that 'under very general conditions' the resulting stress-energy tensor satisfies the energy conditions. This claim is not established. For each detector state g(x), one must find a fluid pressure P and energy density ρ solving Eq. (34) with the relation (35) and satisfying ρ>0, 0≤P/ρ≤1/3, and the energy conditions. No existence or regularity theorem is given for arbitrary g(x); the paper only solves one spherically symmetric example with g=0 and treats one excited state numerically. The claim should either be proved in the required generality or explicitly restricted to the constructed examples.
[Sec. V, Eqs. (45)-(49) and Sec. VI, Eq. (63)] The fluid pressure P(r) is defined only through the integral in Eq. (45), with no closed form. The statements that P is smooth, positive, and decreasing for µ<ℓ², and the energy-condition threshold µ<ℓ²/(1+(1-3η)g0/2), are asserted without proof; the value of P(0) in Eq. (48) is attributed to Mathematica. Similarly, after Eq. (63) the text says 'it is simple to check that all energy conditions are verified' for the total tensor T_0, but no explicit inequalities or parameter ranges are given. The energy-condition verification for the ground-state tensor therefore rests on unproved analytic claims and plots. The authors should supply the missing inequalities or clearly label the verification as numerical for specific parameter values.
[Sec. VI, after Eq. (71)] For the excited state, the components ρ1, R1, and P1 are not displayed; the text says their expressions 'are cumbersome and do not provide any important insight.' Consequently the claimed energy-condition verification for the excited state is entirely plot-based. Since the abstract makes a general claim, the excited-state case needs either explicit expressions with a parameter range over which the energy conditions hold, or a documented numerical verification with the relevant data/code made available.
[Sec. VI, Eq. (54)] The renormalized stress-energy tensor is defined by normal ordering with respect to the detector vacuum |0d⟩. In the presence of a non-trivial confining potential this is a state-dependent subtraction, and normal ordering is not a covariant renormalization scheme. The paper should justify that this prescription gives a well-defined, physically meaningful stress-energy tensor and that the energy-condition results are not artifacts of this renormalization choice.

minor comments (6)

[Sec. VI, after Eq. (71)] The text says that ρ1, R1, and P1 play the same role as ρ0, R0, and P0 in Eq. (4), but the intended reference is Eq. (62).
[Fig. 5 caption] The caption contains a typo: 'η = 0 =, µ = ℓ²/5' should read 'η = 0, µ = ℓ²/5'.
[Sec. IV, after Eq. (35)] The statement that 'this solution is stable' is not supported by any stability analysis; at most the solution is stationary. Please either provide a proof or rephrase.
[Sec. VI, after Eq. (63)] The sentence 'The radial and angular pressures are negative assume negative values' contains a typo and should be revised.
[Sec. V, Eq. (47) and Fig. 3] The equation-of-state parameter is written as w = p/ρ in the text and w := P/ρ in Fig. 3; please unify the notation.
[Sec. VI, Eq. (60)] The function Λ(x) in Eq. (60) includes both the temporal switching and the spatial profile, whereas earlier equations use ζ(x) for the interaction profile. Clarify the relation between these notations.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the stress-energy tensor is derived from a fixed Lagrangian, and the energy-condition checks are explicit calculations rather than built-in constraints.

full rationale

The derivation chain starts from a fixed covariant Lagrangian (Eq. 20) and obtains the stress-energy tensor (Eq. 25) by variation. Conservation is imposed as a constraint (Eq. 27) and solved for the fluid variables ρ and P. The explicit model chooses ψc, Vc, and Lfluid (Eqs. 36, 42) so that Eq. (30) is satisfied; this is a model-building ansatz, not a circular reduction, because the chosen functions are part of the definition of the theory and the subsequent equations of motion are solved, not assumed. The energy-condition verification is an explicit calculation for the constructed example, with the parameter µ restricted to make ρ−|P|>0; the abstract's 'very general conditions' overstates the demonstrated scope, but an overclaim is not a circularity. The only same-author citation ([17]) connects the localized-QFT model to UDW detectors; the stress-energy tensor derivation and the energy-condition checks do not rely on that equivalence, so the self-citation is not load-bearing. No fitted data, no imported uniqueness theorem, and no input redefined as a prediction were found.

Assumptions & free parameters 6 free parameters · 6 assumptions · 1 invented entities

The central construction rests on a handful of hand-chosen parameters (alpha, mu, mc, md, eta, ell) and on the assumption that a perfect fluid can be tuned to an arbitrary on-shell Lagrangian profile while remaining physical. The fluid-field coupling is an invented interaction with no independent evidence. The paper is a designed model rather than a parameter-free derivation.

free parameters (6)

alpha = -6
Coupling between the detector field and the localizing field; chosen as -6 so the effective sech^2 potential supports exactly one bound mode.
mu = ell^2/5 in plots; allowed range 0 < mu < ell^2/(1+g0/2) ~ 0.565 ell^2 for eta=0
Fluid-detector coupling constant; constrained by smoothness of P(r) and by energy conditions. The plotted value is a hand choice.
mc = 2/ell in plots; constrained by mc > 1/ell
Mass of the complex scalar field; must exceed 1/ell so omega_c is real. The plotted value is a convenient choice.
md = 5/ell in Fig. 6
Mass of the detector field; appears in the excited-state calculation. Chosen for the plot.
eta = 0 or 1
Parameter selecting the on-shell fluid Lagrangian (Lfluid = -rho for eta=0; Lfluid = -rho+3P for eta=1). Two model variants are considered.
ell = length scale, sets units
Spatial size of the detector; all other parameters are expressed in units of ell. It is the overall scale of the model.

assumptions (6)

standard math Standard canonical quantization of a scalar field in a confining potential yields the mode structure and Fock space used in Eq. (5).
Used throughout Section II to define the detector field and its states.
standard math The stress-energy tensor is obtained by the Hilbert prescription, variation of the action with respect to the metric, leading to Eq. (25).
Standard in general relativity; used to define the detector's SEM tensor.
domain assumption The perfect fluid is described by T_fluid = (rho+P)u u + P g and satisfies the equations of motion derived from the full action, giving Eq. (27).
The fluid is necessary to localize psi_c; its dynamics are constrained by conservation of the total stress-energy tensor.
domain assumption The backreaction of the quantum field phi_d on the localizing fields is captured by the semiclassical expectation value <:phi_d^2:>, replacing g(x) in the equations.
This assumes fluctuations of phi_d beyond the one-point function do not affect the fluid and psi_c, a semiclassical approximation.
ad hoc to paper The fluid's on-shell Lagrangian can be tuned to the specific profile Lfluid = -2/(mu ell^2) tanh(r/ell)/(r/ell) while the fluid remains a perfect fluid with 0 <= w <= 1/3.
This choice, Eq. (42), is what makes the ansatz for psi_c a solution; no independent physical principle selects this profile.
ad hoc to paper The self-interaction potential Vc(|psi_c|^2) = -(|psi_c|^2)^2 is chosen to realize the example.
This is a model choice, not deduced from a deeper theory.

invented entities (1)

Non-minimal coupling (1 - mu|psi_c|^2) Lfluid between the perfect fluid and the complex scalar field
purpose: This coupling lets the fluid define the spatial profile of psi_c and absorb backreaction from the detector field, keeping the localization stable.
The coupling is introduced in Eq. (20) ad hoc; no independent observational or experimental evidence is given for it.

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Cite this review

Pith. "Pith review of The stress-energy tensor of an Unruh-DeWitt detector." pith.science (2026). https://pith.science/paper/K5M4GY3K

@misc{pith2026241109732,
  author       = {Pith},
  title        = {Pith review of: The stress-energy tensor of an Unruh-DeWitt detector},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K5M4GY3K}},
  note         = {Machine review of arXiv:2411.09732}
}

abstract

We propose a model for a finite-size particle detector, which allows us to derive its stress-energy tensor. This tensor is obtained from a covariant Lagrangian that describes not only the quantum field that models the detector, $\phi_{\text{d}}$, but also the systems responsible for its localization: a complex scalar field, $\psi_{\text{c}}$, and a perfect fluid. The local interaction between the detector and the complex field ensures the square integrability of the detector modes, while the fluid serves to define the spatial profile of $\psi_{\text{c}}$, localizing it in space. We then demonstrate that, under very general conditions, the resulting energy tensor -- incorporating all components of the system -- is physically reasonable and satisfies the energy conditions.

Figures

Figures reproduced from arXiv: 2411.09732 by the authors.

**Figure 1.** ρ + P, ρ + 3P, ρ − |P| as a function of r/ℓ for η = 0 and µ = ℓ 2 /5. cases η = 0 and η = 1. This is specially important for the case η = 1, where the fluid is constituted by particle [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

**Figure 2.** ρ + P, ρ + 3P, ρ − |P| as a function of r/ℓ for η = 1 and µ = ℓ 2 /5 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** w := P/ρ for η = 0 and η = 1 with the choice µ = ℓ 2 /5. with fixed mass and structure. In any case, the fluid is composed of non-exotic matter content. Finally, as expected, both quantities are localized around the origin within a lengthscale characterized by the parameter ℓ. With these classical solutions for the fields ψc(x), and for the fluid parameters ρ(r) and P(r), the field ϕd experiences the effective pot… view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: We see that the radial and angular pressures are negative assume negative values, but it is simple to check that all energy conditions are verified. Notice that these results are independent of any specific property of the field ϕˆ d, as they correspond only to the sys…

**Figure 5.** Figure 5: The pressure deviator Π(r)/p(r) for η = 0 =, µ = ℓ 2 /5, mc = 2 ℓ . The detector’s first excited state is |1d⟩ = ˆa † 1 |0d⟩. To compute the stress-energy tensor when the detector is in this configuration, one could perform the same procedure as that of Section V, usi…

**Figure 7.** Figure 7: The leading order excitation probability of the de [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

**Figure 6.** Figure 6: ρ1(r), R1(r), and P1(r) for η = 0, µ = ℓ 2 /5, mc = 2 ℓ , and md = 5 ℓ . A detector that starts in its ground state evolves according to the unitary time evolution operator Uˆ I = T exp −i Z dV Hˆ I (x) . (72) In the case where the field ϕˆ d starts in the state |0…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Vacuum fluctuations and the renormalized stress-energy tensor on a cone with arbitrary boundary conditions
hep-th 2025-09 conditional novelty 6.0 of 10

A massive scalar on a cone has a stable bound state when M>q, and the paper calculates the renormalized vacuum fluctuations and stress-energy tensor including this bound state.

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K. Bueley, L. Huang, K. Gallock-Yoshimura, and R. B. Mann, Harvesting mutual information from btz black hole spacetime, Phys. Rev. D 106, 025010 (2022)

2022
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D. Mendez-Avalos, L. J. Henderson, K. Gallock- Yoshimura, and R. B. Mann, Entanglement harvesting of three unruh-dewitt detectors, Gen. Relativ. and Gravit. 54, 87 (2022)

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M. Hotta, J. Matsumoto, and G. Yusa, Quantum energy teleportation without a limit of distance, Phys. Rev. A 89, 012311 (2014)

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N. Funai and E. Mart ´ ın-Mart ´ ınez, Engineering negative stress-energy densities with quantum energy teleporta- tion, Phys. Rev. D 96, 025014 (2017)

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N. A. Rodr ´ ıguez-Briones, H. Katiyar, E. Mart ´ ın- Mart ´ ınez, and R. Laflamme, Experimental activation of strong local passive states with quantum information, Phys. Rev. Lett. 130, 110801 (2023)

2023
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R. H. Jonsson, E. Mart ´ ın-Mart ´ ınez, and A. Kempf, In- formation transmission without energy exchange, Phys. Rev. Lett. 114, 110505 (2015)

2015
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Blasco, L

A. Blasco, L. J. Garay, M. Mart ´ ın-Benito, and E. Mart ´ ın- Mart ´ ınez, Timelike information broadcasting in cosmol- ogy, Phys. Rev. D 93, 024055 (2016)

2016
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Blasco, L

A. Blasco, L. J. Garay, M. Mart ´ ın-Benito, and E. Mart ´ ın- Mart ´ ınez, Violation of the strong huygen’s principle and timelike signals from the early universe, Phys. Rev. Lett. 114, 141103 (2015)

2015
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Yamaguchi, A

K. Yamaguchi, A. Ahmadzadegan, P. Simidzija, A. Kempf, and E. Mart ´ ın-Mart ´ ınez, Superadditivity of channel capacity through quantum fields, Phys. Rev. D 101, 105009 (2020)

2020
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A. Ahmadzadegan, P. Simidzija, M. Li, and A. Kempf, Neural networks can learn to utilize correlated auxiliary noise, Sci. Rep. 11, 21624 (2021)

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2022
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T. R. Perche, J. Polo-G´ omez, B. d. S. L. Torres, and E. Mart ´ ın-Mart ´ ınez, Fully relativistic entanglement har- vesting, Phys. Rev. D 109, 045018 (2024)

2024
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2024
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1987

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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2020

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J. Polo-G´ omez, L. J. Garay, and E. Mart ´ ın-Mart ´ ınez, A detector-based measurement theory for quantum field theory, Phys. Rev. D 105, 065003 (2022)

2022

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E. Mart ´ ın-Mart ´ ınez, M. Montero, and M. del Rey, Wavepacket detection with the unruh-dewitt model, Phys. Rev. D 87, 064038 (2013)

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A. Pozas-Kerstjens and E. Mart ´ ın-Mart ´ ınez, Entangle- ment harvesting from the electromagnetic vacuum with hydrogenlike atoms, Phys. Rev. D 94, 064074 (2016)

2016

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Funai, J

N. Funai, J. Louko, and E. Mart ´ ın-Mart ´ ınez,ˆp · ˆA vs ˆx · ˆE: Gauge invariance in quantum optics and quantum field theory, Phys. Rev. D 99, 065014 (2019)

2019